Question

In the adjoining figure, O is the centre of the circle. From point R, seg RM and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm and radius of the circle = 5 cm, then
(1) What is the length of each tangent segment ?
(2) What is the measure of ∠MRO ?
(3) What is the measure of ∠ MRN ?

Answer 1

(1) It is given that seg RM and seg RN are tangent segments touching the circle at M and N, respectively.

∴ ∠OMR = ∠ONR = 90º (Tangent at any point of a circle is perpendicular to the radius throught the point of contact)



OM = 5 cm and OR = 10 cm

In right ∆OMR,

$$\begin{gathered} {\mathrm{OR}}^2={\mathrm{OM}}^2+{\mathrm{MR}}^2 \\ \Rightarrow \mathrm{MR}=\sqrt{{\mathrm{OR}}^2-{\mathrm{OM}}^2} \\ \Rightarrow \mathrm{MR}=\sqrt{{10}^2-5^2} \\ \Rightarrow \mathrm{MR}=\sqrt{100-25}=\sqrt{75}=5\sqrt{3}\mathrm{cm} \end{gathered}$$

Tangent segments drawn from an external point to a circle are congruent.

∴ MR = NR = $5\sqrt{3}$ cm

(2) In right ∆OMR,

$$\begin{gathered} \tan \angle \mathrm{MRO}=\frac{\mathrm{OM}}{\mathrm{MR}} \\ \Rightarrow \tan \angle \mathrm{MRO}=\frac{5\mathrm{cm}}{5\sqrt{3}\mathrm{cm}}=\frac{1}{\sqrt{3}} \\ \Rightarrow \tan \angle \mathrm{MRO}=\tan 30° \\ \Rightarrow \angle \mathrm{MRO}=30° \end{gathered}$$

Thus, the measure of ∠MRO is 30º.

Similarly, ∠NRO = 30º

(3) ∠MRN = ∠MRO + ∠NRO = 30º + 30º = 60º

Thus, the measure of ∠MRN is 60º.